Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honor of Frigyes Riesz.

This article will describe his theorem concerning the dual of a Hilbert space, which is sometimes called the Fréchet–Riesz theorem. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.

The Hilbert space representation theorem

This theorem establishes an important connection between a Hilbert space and its (continuous) dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically antiisomorphic. The (anti-) isomorphism is a particular, natural one as will be described next.

Let H be a Hilbert space, and let H* denote its dual space, consisting of all continuous linear functionals from H into the field $\mathbb {R}$ or $\mathbb {C}$. If $x$ is an element of H, then the function $\varphi _{x}$, for all $y$ in H defined by:

$\varphi _{x}(y)=\left\langle y,x\right\rangle ,$

where $\langle \cdot ,\cdot \rangle$ denotes the inner product of the Hilbert space, is an element of H*. The Riesz representation theorem states that every element of H* can be written uniquely in this form. Given any continuous linear functional g in H*, the corresponding element $x_{g}\in H$ can be constructed uniquely by $x_{g}=g(e_{1})e_{1}+g(e_{2})e_{2}+...$, where $\{e_{i}\}$ is an orthonormal basis of H, and the value of $x_{g}$ does not vary by choice of basis. Thus, if $y\in H,y=a_{1}e_{1}+a_{2}e_{2}+...$, then $g(y)=a_{1}g(e_{1})+a_{2}g(e_{2})+...=\langle x_{g},y\rangle$.

Theorem. The mapping $\Phi$: H → H* defined by $\Phi (x)$ = $\varphi _{x}$ is an isometric (anti-) isomorphism, meaning that:

• $\Phi$ is bijective.
• The norms of $x$ and $\varphi _{x}$ agree: $\Vert x\Vert =\Vert \Phi (x)\Vert$.
• $\Phi$ is additive: $\Phi (x_{1}+x_{2})=\Phi (x_{1})+\Phi (x_{2})$.
• If the base field is $\mathbb {R}$, then $\Phi (\lambda x)=\lambda \Phi (x)$ for all real numbers λ.
• If the base field is $\mathbb {C}$, then $\Phi (\lambda x)={\bar {\lambda }}\Phi (x)$ for all complex numbers λ, where ${\bar {\lambda }}$ denotes the complex conjugation of $\lambda$.

The inverse map of $\Phi$ can be described as follows. Given a non-zero element $\varphi$ of H*, the orthogonal complement of the kernel of $\varphi$ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set $x={\overline {\varphi (z)}}\cdot z/{\left\Vert z\right\Vert }^{2}$. Then $\Phi (x)$ = $\varphi$.

Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).

In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra–ket notation. The theorem says that, every bra $\langle \psi |$ has a corresponding ket $|\psi \rangle$, and the latter is unique.

References

• M. Fréchet (1907). Sur les ensembles de fonctions et les opérations linéaires. C. R. Acad. Sci. Paris 144, 1414–1416.
• F. Riesz (1907). Sur une espèce de géométrie analytique des systèmes de fonctions sommables. C. R. Acad. Sci. Paris 144, 1409–1411.
• F. Riesz (1909). Sur les opérations fonctionnelles linéaires. C. R. Acad. Sci. Paris 149, 974–977.
• P. Halmos Measure Theory, D. van Nostrand and Co., 1950.
• P. Halmos, A Hilbert Space Problem Book, Springer, New York 1982 (problem 3 contains version for vector spaces with coordinate systems).
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
• "Proof of Riesz representation theorem for separable Hilbert spaces". PlanetMath.